Jason Ayto has been acting as a caretaker after Edu's resignation and departure for Nottingham Forest earlier this season.
Arsenal did not sign a senior player over the winter market.
“The decision I think will be done relatively quickly, whether it’s Jason or somebody else I don’t know but I think that decision will be done quite early,” Arteta said.
“We have the sporting director which is Jason, it’s clear and he’s been since the departure of Edu from day one he has been given full responsibility, all our support from that aspect and what the club wants to do after that is not on me.
“It’s a club decision to try to appoint the person that they believe is the best and I will be in full support of that.”
He added on the January transfer window: “Last summer it was the same. We wanted to do certain things, some of them are renewals, some of them try to recruit some players, some with the academy, some others we didn’t want to sell. This window was the same.
“I think Jason and the team have both done a really good job. But then it’s the market, and it’s complicated.”